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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.runs_test"></a><a class="link" href="runs_test.html" title="Runs tests">Runs tests</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.runs_test.h0"></a>
      <span class="phrase"><a name="math_toolkit.runs_test.synopsis"></a></span><a class="link" href="runs_test.html#math_toolkit.runs_test.synopsis">Synopsis</a>
    </h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">runs_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span> <span class="special">{</span>

<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">runs_above_and_below_threshold</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">threshold</span><span class="special">);</span>

<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">runs_above_and_below_median</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>

<span class="special">}}}</span>
</pre>
<h4>
<a name="math_toolkit.runs_test.h1"></a>
      <span class="phrase"><a name="math_toolkit.runs_test.background"></a></span><a class="link" href="runs_test.html#math_toolkit.runs_test.background">Background</a>
    </h4>
<p>
      The "runs above and below median test" tries to determine if a sequence
      is random by looking at the number of consecutive values which exceed (or are
      below) the median of the sequence. For example, if we are given data {5, 2,
      0, 4, 7, 9, 10, 6, 1, 8, 3}, first we find the median (5), and transform the
      vector into a list of + and -'s, depending on whether the value is greater
      or less than the median. (Values equal to the median we simply ignore-this
      is a convention we have inherited from the wonderful <code class="computeroutput"><span class="identifier">randtests</span></code>
      package in R.) Hence our data vector is transformed into
    </p>
<p>
      {-,-,-,+,+,+,-,+,-}
    </p>
<p>
      which is 5 runs, with <span class="emphasis"><em>n</em></span><sub>a</sub> = 5 values above and <span class="emphasis"><em>n</em></span><sub>b</sub> =
      5 values below the median. <a href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda35d.htm" target="_top">NIST</a>
      tells us the expected number of runs and their variance:
    </p>
<p>
      <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/expected_runs_above_threshold.svg" width="418" height="45"></object></span>
    </p>
<p>
      from which we derive the test statistic
    </p>
<p>
      <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/runs_test_statistic.svg" width="115" height="42"></object></span>
    </p>
<p>
      whose distribution we approximate as normal to extract a <span class="emphasis"><em>p</em></span>-value.
    </p>
<p>
      An example usage is as follows:
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">runs_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">{</span><span class="number">5</span><span class="special">,</span> <span class="number">2</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">4</span><span class="special">,</span> <span class="number">7</span><span class="special">,</span> <span class="number">9</span><span class="special">,</span> <span class="number">10</span><span class="special">,</span> <span class="number">6</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">8</span><span class="special">,</span> <span class="number">3</span><span class="special">};</span>
<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">t</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">runs_above_and_below_median</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
<span class="comment">// t =  -0.670820393249936919; p = 0.502334954360502017;</span>
</pre>
<p>
      We see that we have about a 50% chance of seeing this number of runs if the
      null hypothesis of randomness is true, and hence the assumption of randomness
      seems reasonable. As always, the test statistic is the first element of the
      pair, and the <span class="emphasis"><em>p</em></span>-value is the second element.
    </p>
<h4>
<a name="math_toolkit.runs_test.h2"></a>
      <span class="phrase"><a name="math_toolkit.runs_test.performance"></a></span><a class="link" href="runs_test.html#math_toolkit.runs_test.performance">Performance</a>
    </h4>
<p>
      There are two cases: Where the threshold (typically the median) has already
      been computed, and the case where the mean and sample variance must be computed
      on the fly. Computing the median is fairly expensive (requiring a call to
      <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">median</span></code>),
      and since the order of the original data must be preserved, it must allocate.
      If you believe your data to come from a distribution where the means and median
      coincide, or if you've already computed the median in the course of some other
      analysis, then you can get away with a call to <code class="computeroutput"><span class="identifier">runs_above_and_below_threshold</span></code>
      via
    </p>
<pre class="programlisting"><span class="identifier">Real</span> <span class="identifier">threshold</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">t</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">runs_above_and_below_threshold</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">threshold</span><span class="special">);</span>
</pre>
<p>
      The performance differences between these two cases are obvious:
    </p>
<pre class="programlisting"><span class="special">---------------------------------------------------------------------------------------</span>
<span class="identifier">Benchmark</span>                                            <span class="identifier">Time</span>   <span class="identifier">Bytes</span><span class="special">/</span><span class="identifier">second</span>
<span class="special">---------------------------------------------------------------------------------------</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8</span>                   <span class="number">260</span> <span class="identifier">ns</span> <span class="number">118.421</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16</span>                  <span class="number">318</span> <span class="identifier">ns</span> <span class="number">192.797</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32</span>                  <span class="number">417</span> <span class="identifier">ns</span> <span class="number">303.509</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">64</span>                  <span class="number">625</span> <span class="identifier">ns</span> <span class="number">390.578</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">128</span>                 <span class="number">743</span> <span class="identifier">ns</span> <span class="number">657.827</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">256</span>                <span class="number">1308</span> <span class="identifier">ns</span> <span class="number">767.181</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">512</span>                <span class="number">1896</span> <span class="identifier">ns</span> <span class="number">1034.31</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1024</span>               <span class="number">6582</span> <span class="identifier">ns</span> <span class="number">594.458</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2048</span>              <span class="number">26067</span> <span class="identifier">ns</span> <span class="number">300.001</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4096</span>              <span class="number">62023</span> <span class="identifier">ns</span> <span class="number">252.125</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8192</span>             <span class="number">124976</span> <span class="identifier">ns</span> <span class="number">250.256</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16384</span>            <span class="number">242171</span> <span class="identifier">ns</span> <span class="number">258.29</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32768</span>            <span class="number">528683</span> <span class="identifier">ns</span> <span class="number">236.714</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">65536</span>            <span class="number">965354</span> <span class="identifier">ns</span> <span class="number">259.185</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">131072</span>          <span class="number">2514693</span> <span class="identifier">ns</span> <span class="number">199.068</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">262144</span>          <span class="number">4223084</span> <span class="identifier">ns</span> <span class="number">237.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">524288</span>          <span class="number">8638963</span> <span class="identifier">ns</span> <span class="number">231.755</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1048576</span>        <span class="number">16215682</span> <span class="identifier">ns</span> <span class="number">246.995</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2097152</span>        <span class="number">39180496</span> <span class="identifier">ns</span> <span class="number">204.443</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4194304</span>        <span class="number">82495779</span> <span class="identifier">ns</span> <span class="number">194.307</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8388608</span>       <span class="number">142675936</span> <span class="identifier">ns</span> <span class="number">224.547</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16777216</span>      <span class="number">287638068</span> <span class="identifier">ns</span> <span class="number">223.088</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span><span class="identifier">_BigO</span>              <span class="number">17.25</span> <span class="identifier">N</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>                  <span class="number">191</span> <span class="identifier">ns</span> <span class="number">320.129</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>                 <span class="number">233</span> <span class="identifier">ns</span> <span class="number">523.526</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>                 <span class="number">334</span> <span class="identifier">ns</span> <span class="number">730.8</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>                 <span class="number">456</span> <span class="identifier">ns</span> <span class="number">1070.93</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>                <span class="number">688</span> <span class="identifier">ns</span> <span class="number">1.38789</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>               <span class="number">1257</span> <span class="identifier">ns</span> <span class="number">1.51807</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>               <span class="number">2663</span> <span class="identifier">ns</span> <span class="number">1.43406</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>              <span class="number">4100</span> <span class="identifier">ns</span> <span class="number">1.86266</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>             <span class="number">23493</span> <span class="identifier">ns</span> <span class="number">665.851</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>             <span class="number">57968</span> <span class="identifier">ns</span> <span class="number">539.551</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>            <span class="number">142272</span> <span class="identifier">ns</span> <span class="number">439.746</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>           <span class="number">260948</span> <span class="identifier">ns</span> <span class="number">479.639</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>           <span class="number">551577</span> <span class="identifier">ns</span> <span class="number">453.623</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>          <span class="number">1056583</span> <span class="identifier">ns</span> <span class="number">473.654</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>         <span class="number">2123956</span> <span class="identifier">ns</span> <span class="number">471.35</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>         <span class="number">5028745</span> <span class="identifier">ns</span> <span class="number">398.111</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>        <span class="number">10399212</span> <span class="identifier">ns</span> <span class="number">384.981</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>       <span class="number">23089767</span> <span class="identifier">ns</span> <span class="number">348.496</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>       <span class="number">37626884</span> <span class="identifier">ns</span> <span class="number">425.962</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>       <span class="number">79281747</span> <span class="identifier">ns</span> <span class="number">404.088</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>      <span class="number">172055781</span> <span class="identifier">ns</span> <span class="number">373.391</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>     <span class="number">391377449</span> <span class="identifier">ns</span> <span class="number">332.01</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowMedian</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>             <span class="number">22.52</span> <span class="identifier">N</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">41.6</span> <span class="identifier">ns</span> <span class="number">739.55</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16</span>              <span class="number">58.4</span> <span class="identifier">ns</span> <span class="number">1050.48</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32</span>              <span class="number">66.5</span> <span class="identifier">ns</span> <span class="number">1.79606</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">64</span>               <span class="number">115</span> <span class="identifier">ns</span> <span class="number">2.0762</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">128</span>              <span class="number">198</span> <span class="identifier">ns</span> <span class="number">2.41515</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">256</span>              <span class="number">365</span> <span class="identifier">ns</span> <span class="number">2.61328</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">512</span>              <span class="number">720</span> <span class="identifier">ns</span> <span class="number">2.65053</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1024</span>            <span class="number">1424</span> <span class="identifier">ns</span> <span class="number">2.68123</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2048</span>            <span class="number">3009</span> <span class="identifier">ns</span> <span class="number">2.5379</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4096</span>           <span class="number">16748</span> <span class="identifier">ns</span> <span class="number">933.699</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8192</span>           <span class="number">40190</span> <span class="identifier">ns</span> <span class="number">778.105</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16384</span>          <span class="number">86500</span> <span class="identifier">ns</span> <span class="number">723.067</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32768</span>         <span class="number">176692</span> <span class="identifier">ns</span> <span class="number">708.108</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">65536</span>         <span class="number">356863</span> <span class="identifier">ns</span> <span class="number">701.198</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">131072</span>        <span class="number">714807</span> <span class="identifier">ns</span> <span class="number">700.08</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">262144</span>       <span class="number">1429078</span> <span class="identifier">ns</span> <span class="number">700.415</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">524288</span>       <span class="number">2877227</span> <span class="identifier">ns</span> <span class="number">695.785</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1048576</span>      <span class="number">5795662</span> <span class="identifier">ns</span> <span class="number">691.222</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2097152</span>     <span class="number">11562715</span> <span class="identifier">ns</span> <span class="number">692.427</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4194304</span>     <span class="number">23364846</span> <span class="identifier">ns</span> <span class="number">686.464</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8388608</span>     <span class="number">46442540</span> <span class="identifier">ns</span> <span class="number">689.871</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16777216</span>    <span class="number">92284501</span> <span class="identifier">ns</span> <span class="number">694.006</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span><span class="identifier">_BigO</span>            <span class="number">5.51</span> <span class="identifier">N</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>              <span class="number">45.1</span> <span class="identifier">ns</span> <span class="number">1.32169</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>             <span class="number">53.6</span> <span class="identifier">ns</span> <span class="number">2.22712</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>             <span class="number">71.4</span> <span class="identifier">ns</span> <span class="number">3.34079</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">112</span> <span class="identifier">ns</span> <span class="number">4.24946</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>             <span class="number">196</span> <span class="identifier">ns</span> <span class="number">4.87317</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>             <span class="number">378</span> <span class="identifier">ns</span> <span class="number">5.04476</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>             <span class="number">702</span> <span class="identifier">ns</span> <span class="number">5.44134</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>           <span class="number">1417</span> <span class="identifier">ns</span> <span class="number">5.3865</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>           <span class="number">3031</span> <span class="identifier">ns</span> <span class="number">5.03872</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>          <span class="number">16813</span> <span class="identifier">ns</span> <span class="number">1.81669</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>          <span class="number">41182</span> <span class="identifier">ns</span> <span class="number">1.48565</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>         <span class="number">86939</span> <span class="identifier">ns</span> <span class="number">1.40536</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>        <span class="number">177255</span> <span class="identifier">ns</span> <span class="number">1.37892</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>        <span class="number">356391</span> <span class="identifier">ns</span> <span class="number">1.3713</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>       <span class="number">718417</span> <span class="identifier">ns</span> <span class="number">1.36057</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>      <span class="number">1442288</span> <span class="identifier">ns</span> <span class="number">1.35583</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>      <span class="number">2942259</span> <span class="identifier">ns</span> <span class="number">1.33217</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>     <span class="number">5870235</span> <span class="identifier">ns</span> <span class="number">1.33244</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>    <span class="number">11743081</span> <span class="identifier">ns</span> <span class="number">1.33192</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>    <span class="number">23521002</span> <span class="identifier">ns</span> <span class="number">1.32976</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>    <span class="number">46917407</span> <span class="identifier">ns</span> <span class="number">1.33339</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>   <span class="number">93823876</span> <span class="identifier">ns</span> <span class="number">1.33305</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">BMRunsAboveAndBelowThreshold</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>           <span class="number">5.59</span> <span class="identifier">N</span>          <span class="number">5.59</span> <span class="identifier">N</span>
</pre>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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